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An automorphism of a spherical building is called domestic if it maps no chamber to an opposite chamber. In this paper we classify domestic automorphisms of spherical buildings of classical type. [ABSTRACT FROM AUTHOR]

The compact 16-dimensional Moufang plane, also known as the Cayley plane, has traditionally been defined through the lens of octonionic geometry. In this study, we present a novel approach, demonstrating that the Cayley plane can be defined in an equally clean, straightforward and more economic way using two different division and composition algebras: the paraoctonions and the Okubo algebra. The result is quite surprising since paraoctonions and Okubo algebra possess a weaker algebraic structure than the octonions, since they are non-alternative and do not satisfy the Moufang identities. Intriguingly, the real Okubo algebra has SU 3 as automorphism group, which is a classical Lie group, while octonions and paraoctonions have an exceptional Lie group of type G 2 . This is remarkable, given that the projective plane defined over the real Okubo algebra is nevertheless isomorphic and isometric to the octonionic projective plane which is at the very heart of the geometric realisations of all types of exceptional Lie groups. Despite its historical ties with octonionic geometry, our research underscores the real Okubo algebra as the weakest algebraic structure allowing the definition of the compact 16-dimensional Moufang plane. [ABSTRACT FROM AUTHOR]

In this paper, we classify the m-ovoids of finite classical polar spaces that admit a transitive automorphism group acting irreducibly on the ambient vector space. In particular, we obtain several new infinite families of transitive m-ovoids. [ABSTRACT FROM AUTHOR]

In Quantum Information theory, graph states are quantum states defined by graphs. In this work we exhibit a correspondence between orbits of graph states and orbits in the variety of binary symmetric principal minors, under the action of SL (2 , F 2) × n ⋊ S n . First we study the orbits of maximal abelian subgroups of the n-fold Pauli group under the action of C n loc ⋊ S n , where C n loc is the n-fold local Clifford group, and we show that this action corresponds to the natural action of SL (2 , F 2) × n ⋊ S n on the variety Z n ⊂ P (F 2 2 n) of principal minors of binary symmetric n × n matrices: the crucial step is in translating the action of SL (2 , F 2) × n into an action of the local symplectic group Sp 2 n loc (F 2) . We conclude by showing how the former action restricts onto stabilizer groups, stabilizer states and graph states. [ABSTRACT FROM AUTHOR]

We review the impact of Jacques Tits' paper, "Ovoïdes et groupes de Suzuki," Arch. Math.13 (1962), 187–198. The paper turned out to be a milestone for both geometry (incidence geometry, finite geometry, Galois geometry) and algebra (group theory). [ABSTRACT FROM AUTHOR]

The relationship of discrete L-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete L-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete L-algebra X is determined and shown to be a complete invariant. It is proved that X ∖ {1} is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality n > 3, a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated L-algebra is determined. [ABSTRACT FROM AUTHOR]

A polar space 풮 is called symplectic if it admits a projective embedding ε : 풮 → PG(V) such that the image ε(풮) of 풮 by ε is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when 풮 admits different (non-isomorphic) embeddings, as it is the case when 풮 is defined over a field of characteristic 2. [ABSTRACT FROM AUTHOR]

We classify symplectic 4-dimensional semifields over F q , for q ≤ 9 , thereby extending (and confirming) the previously obtained classifications for q ≤ 7 . The classification is obtained by classifying all symplectic semifield subspaces in PG (9 , q) for q ≤ 9 up to K-equivalence, where K ≤ PGL (10 , q) is the lift of PGL (4 , q) under the Veronese embedding of PG (3 , q) in PG (9 , q) of degree two. Our results imply the non-existence of non-associative symplectic 4-dimensional semifields for q even, q ≤ 8 . For q odd, and q ≤ 9 , our results imply that the isotopism class of a symplectic non-associative 4-dimensional semifield over F q is contained in the Knuth orbit of a Dickson commutative semifield. [ABSTRACT FROM AUTHOR]

In this paper we compute the generating rank of k-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of k-Grassmannians arising from Hermitian forms of Witt index n defined over vector spaces of dimension N > 2n. We also study generating sets for the 2-Grassmannians arising from quadratic forms of Witt index n defined over V(N, 𝔽q) for q = 4, 8, 9 and 2n ≤ N ≤ 2n + 2. We prove that for N > 6 and anisotropic defect (polar corank) d ≠ 2 they can be generated over the prime subfield, thus determining their generating rank. [ABSTRACT FROM AUTHOR]

In this paper, we complete the classification of transitive ovoids of finite Hermitian polar spaces. [ABSTRACT FROM AUTHOR]

We consider Grassmann structures defined on the family consisting of subspaces on which a given nondegenerate bilinear form defined on a real vector space is positive definite. One may call such structures Grassmann spaces over generalized hyperbolic spaces. We show that the underlying (generalized) hyperbolic space can be recovered in terms of its Grassmannian, and the underlying projective space (equipped with respective associated polarity) can be recovered in terms of the generalized hyperbolic space defined over it. [ABSTRACT FROM AUTHOR]

One of the most fundamental results in the theory of regular near polygons is the result that every regular near 2d-gon, d ≥ 3 , whose parameters s , t , t i , i ∈ { 0 , 1 , ... , d } , satisfy s , t 2 ≥ 2 and t 3 = t 2 2 + t 2 is a dual polar space. The proof of that theorem heavily relies on Tits' theory of buildings, in particular on Tits' strong results on covering of chamber systems. In this paper, we give an alternative proof which only employs geometrical and algebraic combinatorial arguments. [ABSTRACT FROM AUTHOR]

In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a strongly regular graph Γ with parameters (216, 40, 4, 8) and proved that it is the unique PSU (4 , 2) -invariant vertex-transitive graph on 216 vertices. In this paper, using the geometry of the Hermitian surface of PG (3 , 4) , we provide a computer-free proof of the existence of the graph Γ . The maximal cliques of Γ are also determined. [ABSTRACT FROM AUTHOR]

An m-cover of lines of a finite projective space PG (r , q) (of a finite polar space P ) is a set of lines L of PG (r , q) (of P ) such that every point of PG (r , q) (of P ) contains m lines of L , for some m. Embed PG (r , q) in PG (r , q 2) . Let L ¯ denote the set of points of PG (r , q 2) lying on the extended lines of L . An m-cover L of PG (r , q) is an (r - 2) -dual m-cover if there are two possibilities for the number of lines of L contained in an (r - 2) -space of PG (r , q) . Basing on this notion, we characterize m-covers L of PG (r , q) such that L ¯ is a two-character set of PG (r , q 2) . In particular, we show that if L is invariant under a Singer cyclic group of PG (r , q) then it is an (r - 2) -dual m-cover. Assuming that the lines of L are lines of a symplectic polar space W (r , q) (of an orthogonal polar space Q (r , q) of parabolic type), similarly to the projective case we introduce the notion of an (r - 2) -dual m-cover of symplectic type (of parabolic type). We prove that an m-cover L of W (r , q) (of Q (r , q) ) has this dual property if and only if L ¯ is a tight set of an Hermitian variety H (r , q 2) or of W (r , q 2) (of H (r , q 2) or of Q (r , q 2) ). We also provide some interesting examples of (4 n - 3) -dual m-covers of symplectic type of W (4 n - 1 , q) . [ABSTRACT FROM AUTHOR]

We construct a homogeneous full projective embedding of the dual polar space DW(2n-1,2) from the hyperplane intersections of hyperbolic type of the parabolic quadric Q(2n, 2). We believe that this embedding is universal, but have not succeeded in proving that. As a by-product of our investigations, we have obtained necessary and sufficient conditions for this to be the case and came across two other homogeneous full projective embeddings of DW(2n-1,2), one with vector dimension 22n-1+3·2n-1-23 and another one with vector dimension 22n-1+3·2n-1-2-6n3. [ABSTRACT FROM AUTHOR]

In this paper we determine the minimum distance of orthogonal line-Grassmann codes for q even. The case q odd was solved in [3] . For n ≠ 3 we also determine their second smallest distance. Furthermore, we show that for q even all minimum weight codewords are equivalent and that the symplectic line-Grassmann codes are proper subcodes of codimension 2 n of the orthogonal ones. [ABSTRACT FROM AUTHOR]

We characterize the valuations of the near polygon Hn that are induced by classical valuations of the dual polar space DW(2n-1,2) into which it is isometrically embeddable. An application to near 2n-gons that contain Hn as a full subgeometry is given. [ABSTRACT FROM AUTHOR]

We define optimal EKR-sets in finite buildings. This definition is motivated by various contributions on optimal EKR-sets in finite projective spaces and polar spaces. Our main result is the classification of optimal EKR-sets of type {ℓ,ℓ-1} in finite building of type Dℓ with ℓ even. As it is the case for most of the known optimal EKR-sets in finite buildings, our EKR-sets have a natural center. This provides some evidence that the EKR-problem for finite buildings and Tits center conjecture are closely related. [ABSTRACT FROM AUTHOR]

Let ( A , B ) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum ( A _ _ , B _ _ ) ⊕ ( A 1 , B 1 ) ⊕ … ⊕ ( A t , B t ) that is congruent to ( A , B ) , in which ( A _ _ , B _ _ ) is a pair of nonsingular matrices and ( A 1 , B 1 ) , … , ( A t , B t ) are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of ( A , B ) under congruence over an algebraically closed field of characteristic not 2. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic not 2, and let ( A , B ) be a pair of n × n matrices over F , in which A is symmetric and B is skew-symmetric. A canonical form of ( A , B ) with respect to congruence transformations ( S T A S , S T B S ) was given by Sergeichuk (1988) [25] up to classification of symmetric and Hermitian forms over finite extensions of F . We obtain a simpler canonical form of ( A , B ) if B is nonsingular. Such a pair ( A , B ) defines a quadratic form on a symplectic space, that is, on a vector space with scalar product given by a nonsingular skew-symmetric form. As an application, we obtain known canonical matrices of quadratic forms and Hamiltonian operators on real and complex symplectic spaces. [ABSTRACT FROM AUTHOR]

By applying a method of Godsil and McKay to some graphs related to the symplectic graph, two series of new infinite families of switched strongly regular graphs with parameters $$\big (2^n\pm 2^{\frac{n-1}{2}},2^{n-1}\pm 2^{\frac{n-1}{2}},2^{n-2} \pm 2^{\frac{n-3}{2}},2^{n-2}\pm 2^{\frac{n-1}{2}}\big )$$ are constructed for $$n \ge 5$$ , where n is odd. The construction is described in terms of the geometry of quadrics in the projective space. The binary linear codes of these switched graphs have parameters $$\big [2^n \mp 2^{\frac{n-1}{2}},n+3,2^{t+1}\big ]_2$$ and $$\big [2^n \mp 2^{\frac{n-1}{2}},n+3,2^{t+2}\big ]_2$$ respectively. [ABSTRACT FROM AUTHOR]

If s is an even power of a prime and t equals s or $$s\sqrt{s}$$ , we show the existence of pseudo-geometric strongly regular graphs having the same parameters of the point-graph of a GQ( s, t), that are not geometric. [ABSTRACT FROM AUTHOR]

We describe isometric embeddings of polar Grassmann graphs formed by non-maximal singular subspaces. In almost all cases, they are induced by collinearity-preserving injections of polar spaces. As a simple consequence of this result, we get a metric characterization of apartments in polar Grassmannians. [ABSTRACT FROM AUTHOR]

We provide new proofs for the non-existence of ovoids in hyperbolic spaces of rank at least four in even characteristic, and for the Hermitian polar space H(5,4). We also improve the results of A. Klein on the non-existence of ovoids of Hermitian spaces and hyperbolic quadrics. [ABSTRACT FROM AUTHOR]

We use Müller and Nagy's method of contradicting subsets to give a new proof for the non-existence of sharply 2-transitive subsets of the symplectic groups $$\mathrm {Sp}(2d,2)$$ in their doubly-transitive actions of degrees $$2^{2d-1}\pm 2^{d-1}$$ . The original proof by Grundhöfer and Müller was rather complicated and used some results from modular representation theory, whereas our new proof requires only simple counting arguments. [ABSTRACT FROM AUTHOR]

It seems to be of recurring interest in the literature to give alternative proofs for the fact that the determinant of a symplectic matrix is one. We state four short and elementary proofs for symplectic matrices over general fields. Two of them seem to be new. [ABSTRACT FROM AUTHOR]

We describe several families of regular partitions of half-spin geometries and determine their associated parameters and eigenvalues. We also give a general method for computing the eigenvalues of regular partitions of half-spin geometries. [ABSTRACT FROM AUTHOR]

Small tight sets have been classified in finite classical polar spaces of symplectic type, hyperbolic type and one of the two Hermitian types [1]. The approach used is not working for the remaining finite classical polar spaces, mainly due to the fact that a tight set for these is not necessarily a minihyper, so the known results on blocking sets can not be applied. This might be the reason that there is no classification for small tight sets in these spaces. This paper provides a classification of tight sets with parameter x in these spaces provided that x is small compared to the order q of the polar space. One of the results is that a tight set of the generalized quadrangle Q(4, q) that is not the union of disjoint lines has at least ( $$\sqrt q $$ + 1)( q + 1) points with equality if and only if it consists of the points of an embedded subquadrangle of order $$\sqrt q $$ . [ABSTRACT FROM AUTHOR]

Given a finite dimensional vector spaceVover a fieldand a non-trivial but possibly degenerate reflexive sesquilinear form, letP(f) be the polar space associated tof, namely the collection of all totallyf-isotropic subspaces ofV, andR(f) the collection of all subspaces ofVthat meetnon-trivially. Letrbe the Witt index off. Given a positive integer, letbe thek-Grassmannian ofV, namely the collection of allk-subspaces ofV, and putand. Letbe the Plücker embedding, mapping a subspaceonto the point. The imageofviais a well-known projective variety, called a Grassmann variety. Putand. In this paper, we find a tensor equation that, jointly with the equations of, describeas a subset of. The last two sections of the paper are devoted to the following two special cases: (1)fis bilinear and; (2)fis alternating. [ABSTRACT FROM PUBLISHER]

In this paper we introduce a set of sufficient criteria for the construction of relative hemisystems of the Hermitian space $${\mathrm {H}}(3,q^2)$$ , unifying all known infinite families. We use these conditions to provide new proofs of the existence of the known infinite families of relative hemisystems. Reproving these results has allowed us to find new relative hemisystems closely related to an infinite family of Cossidente's, and develop techniques that are likely to be useful in finding relative hemisystems in future. [ABSTRACT FROM AUTHOR]

In (Bull Belg Math Soc Simon Stevin 4:299-316, 1997), Shult introduced a class of parapolar spaces, the so-called near-polar spaces. We introduce here the notion of a polarized non-abelian representation of a slim near-polar space, that is, a near-polar space in which every line is incident with precisely three points. For such a polarized non-abelian representation, we study the structure of the corresponding representation group, enabling us to generalize several of the results obtained in Sahoo and Sastry (J Algebraic Comb 29:195-213, 2009) for non-abelian representations of slim dense near hexagons. We show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding. [ABSTRACT FROM AUTHOR]

In this paper we show that n-dimensional dual hyperovals cannot exist in all but one classical polar space of rank n if n is even. This resolves a question posed by Yoshiara. [ABSTRACT FROM AUTHOR]

Fundamental theorem of geometry of hermitian matrices characterizes all bijective maps on the space of all hermitian matrices, which preserve adjacency in both directions. In this and subsequent paper we characterize maps on the set of all invertible hermitian matrices over a finite field, which preserve adjacency in one direction. In this first paper it is shown that maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are necessarily bijective, so the corresponding graph on invertible hermitian matrices, where edges are defined by the adjacency relation, is a core. Besides matrix theory, the proof relies on results from several other mathematical areas, including spectral and chromatic graph theory, and finite geometry. [ABSTRACT FROM AUTHOR]

The structure of lines in a projective or a polar Grassmann space together with binary coplanarity is a sufficient system of primitive notions for these geometries. [ABSTRACT FROM AUTHOR]

Let $${\mathbb F}$$ and $${\mathbb F}'$$ be two fields such that $${\mathbb F}'$$ is a quadratic Galois extension of $${\mathbb F}$$ . If $$|{\mathbb F}| \ge 3$$ , then we provide sufficient conditions for a hyperplane of the Hermitian dual polar space $$DH(5,{\mathbb F}')$$ to arise from the Grassmann embedding. We use this to give an alternative proof for the fact that all hyperplanes of $$DH(5,q^2)$$ , $$q \not = 2$$ , arise from the Grassmann embedding, and to show that every hyperplane of $$DH(5,{\mathbb F}')$$ that contains a quad $$Q$$ is either classical or the extension of a non-classical ovoid of $$Q$$ . We will also give a classification of the hyperplanes of $$DH(5,{\mathbb F}')$$ that contain a quad and arise from the Grassmann embedding. [ABSTRACT FROM AUTHOR]

Polar Grassmann codes of orthogonal type have been introduced in [1] . They are punctured versions of the Grassmann code arising from the projective system defined by the Plücker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann codes of orthogonal type for q odd. [ABSTRACT FROM AUTHOR]

We provide a combinatorial characterization of LG(3,6)(${\mathbb{K}}$) using an axiom set which is the natural continuation of the Mazzocca–Melone set we used to characterize Severi varieties over arbitrary fields (Schillewaert and Van Maldeghem, Severi varieties over arbitrary fields, Preprint). This fits within a large project aiming at constructing and characterizing the varieties related to the Freudenthal–Tits magic square. [ABSTRACT FROM AUTHOR]

Consider a finite classical polar space of rank $$d\ge 2$$ and an integer n with $$0

In this paper we introduce symplectic Grassmann codes, in analogy to ordinary Grassmann codes and orthogonal Grassmann codes, as projective codes defined by symplectic Grassmannians. Lagrangian–Grassmannian codes are a special class of symplectic Grassmann codes. We describe all the parameters of line symplectic Grassmann codes and we provide the full weight enumerator for the Lagrangian–Grassmannian codes of rank 2 and 3. [ABSTRACT FROM AUTHOR]

We establish that every embedding of a Grassmann graph in a polar Grassmann graph can be reduced to an embedding in a Grassmann graph or to an embedding in the collinearity graph of a polar space. Also, we consider 3-embeddings, i.e. embeddings preserving all distances not greater than 3, of dual polar graphs whose diameter is not less than 3 in polar Grassmann graphs formed by non-maximal singular subspaces. Using the same arguments we show that every such an embedding can be reduced to an embedding in a Grassmann graph. [ABSTRACT FROM AUTHOR]

We study the geometry of involutions of the finite unitary group P Γ U ( n + 1 , q 2 ) . The configurations of fixed singular points, fixed singular lines and fixed symplectic subgeometries in the case n = 3 , are provided. This allows us to describe the geometry of linear involutions of PGO − ( 6 , q ) . [ABSTRACT FROM AUTHOR]